Properties of fuzzy set helps us to simplify many mathematical fuzzy set operations. Sets are collection of unordered, district elements. We can perform various fuzzy set operations on the fuzzy set. It is recommended to reader to first navigate through the fuzzy set operations for better understanding of properties of fuzzy set.

Most of the properties of crisp sets are hold for fuzzy set also.

Note: To differentiate fuzzy sets from the classical set, we will be putting bar under or over the set notation

Properties of Fuzzy Sets:

Involution

Involution states that the complement of complement is set it self.

( A‘ )’ = A

Commutativity

Operations are called commutative if the order of operands does not alter the result. Fuzzy sets are commutative under union and intersection operations.

AB = BA

AB = BA

Associativity

Associativity allows to change the order of operations performed on operand, how ever relative order of operand can not be changed. All sets in equation must appear in the identical order only. Fuzzy sets are associative under union and intersection operations.

A ∪ ( BC ) = ( AB ) ∪ C

A ∩ ( BC ) = ( AB ) ∩ C

Distributivity

A ∪ ( BC ) = ( AB ) ∩ ( AC )

A ∩ ( BC ) = ( AB ) ∪ ( AC )

Absorption

Absorption produces the identical sets after stated union and intersection operations.

A ∪ ( AB) = A

A ∩ ( AB ) = A

Idempotency / Tautology

Idempotency does not alter the element or the membership value of elements in the set

AA = A

AA = A

Identity

A ∪ ϕ = A

A ∩ ϕ = ϕ

A ∪ X = X

A ∩ X = A

Transitivity

If AB and BC then AC

De Morgan’s Law

De Morgan’s Laws can be stated as,

  • The complement of a union is the intersection of the complement of individual sets
  • The complement of a intersection is the union of the complement of individual sets

( AB )’ = A‘ ∩ B

( AB )’ = A’B

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Let a = μA(x) and b = μB(x), we can define the properties of following operations as,

Fuzzy Complement

C:[0, 1]→[0, 1], which satisfies the following axioms

  • Axiom 1: C(0) = 1, C(1) = 0 (boundary condition)
  • Axiom 2: If a < b, then c(a) ≥ c(b)
  • Axiom 3: C is continuous
  • Axiom 4: C(C(a)) = a

Axiom 1 and Axiom 2 forms Axiomatic Skeleton for fuzzy complement

Fuzzy Union

U:[0, 1]×[0, 1]→[0, 1]

  • Axiom 1: U(0, 0)=0, U(1, 0)=1, U(0, 1)=1, U(1, 1)=1   (Boundary Condition)
  • Axiom 2: If a < a′ and b < b′ then U(a, b) ≤ U(a′, b′)  (monotonic)
  • Axiom 3: Commutative: U(a, b) = U(b, a)
  • Axiom 4: Associative: U(U(a, b), c) = U(a, U(b, c))
  • Axiom 5: U is continuous
  • Axiom 6: U(a, a) = a  (Idempotency)

Axiom 1 to 4 form Axiomatic Skeleton for fuzzy union

Fuzzy Intersection

I:[0, 1]×[0, 1]→[0, 1]

  • Axiom 1: I(0, 0)=0, I(1, 0)=0, I(0, 1)=0, I(1, 1)=1   (Boundary Condition)
  • Axiom 2: If a<a′ and b<b′ then I(a, b) ≤ I(a′, b′ )    (monotonic)
  • Axiom 3: Commutative: I(a, b) = I(b, a)
  • Axiom 4: Associative: I(I(a, b), c) = I(a, I(b, c))
  • Axiom 5: I is continuous
  • Axiom 6: I(a, a) = a  (Idempotency)

Axiom 1 to 4 form Axiomatic Skeleton for fuzzy intersection